Asymptotic properties of the least-squares method for estimating transfer functions and disturbance spectra

1992 ◽  
Vol 24 (02) ◽  
pp. 412-440 ◽  
Author(s):  
Lennart Ljung ◽  
Bo Wahlberg
Keyword(s):  
Transfer Function ◽  
Finite Order ◽  
Transfer Functions ◽  
Least Squares Method ◽  
Asymptotic Properties ◽  
Model Order ◽  
Asymptotically Normal ◽  
Strongly Consistent ◽  
True System ◽  
Asymptotic Variances

The problem of estimating the transfer function of a linear system, together with the spectral density of an additive disturbance, is considered. The set of models used consists of linear rational transfer functions and the spectral densities are estimated from a finite-order autoregressive disturbance description. The true system and disturbance spectrum are, however, not necessarily of finite order. We investigate the properties of the estimates obtained as the number of observations tends to ∞ at the same time as the model order employed tends to ∞. It is shown that the estimates are strongly consistent and asymptotically normal, and an expression for the asymptotic variances is also given. The variance of the transfer function estimate at a certain frequency is related to the signal/noise ratio at that frequency and the model orders used, as well as the number of observations. The variance of the noise spectral estimate relates in a similar way to the squared value of the true spectrum.

Asymptotic properties of the least-squares method for estimating transfer functions and disturbance spectra

1992 ◽  
Vol 24 (2) ◽  
pp. 412-440 ◽  
Author(s):  
Lennart Ljung ◽  
Bo Wahlberg
Keyword(s):  
Transfer Function ◽  
Finite Order ◽  
Transfer Functions ◽  
Least Squares Method ◽  
Asymptotic Properties ◽  
Model Order ◽  
Asymptotically Normal ◽  
Strongly Consistent ◽  
True System ◽  
Asymptotic Variances

The problem of estimating the transfer function of a linear system, together with the spectral density of an additive disturbance, is considered. The set of models used consists of linear rational transfer functions and the spectral densities are estimated from a finite-order autoregressive disturbance description. The true system and disturbance spectrum are, however, not necessarily of finite order. We investigate the properties of the estimates obtained as the number of observations tends to ∞ at the same time as the model order employed tends to ∞. It is shown that the estimates are strongly consistent and asymptotically normal, and an expression for the asymptotic variances is also given. The variance of the transfer function estimate at a certain frequency is related to the signal/noise ratio at that frequency and the model orders used, as well as the number of observations. The variance of the noise spectral estimate relates in a similar way to the squared value of the true spectrum.

scholarly journals Behavioral Analysis of Various Techniques of Model Order Reduction Used in the Reduction of Large Scale Control System

2019 ◽  
Vol 9 (2) ◽  
pp. 4894-4899
Keyword(s):  
Transfer Function ◽  
Model Order Reduction ◽  
Large Scale ◽  
Transfer Functions ◽  
Order Approximation ◽  
Order Reduction ◽  
Higher Order ◽  
Reduction Technique ◽  
Order System ◽  
Model Order

It is very important task to study the behavior of the processes occurring in the industry. To attain this task, the knowledge of the transfer function of the system should be there. When working in robust environment, these transfer functions becomes so tedious that it becomes very difficult to obtain these transfer functions and hence affects the study of the behavior of these system. Due to this, the requirement for reduction of these transfer function becomes a necessity to analyze the behavior of foresaid systems and it becomes easy to do the desired modifications in the system i.e addition of any feature, desired changes in the behavior etc., furthermore the thing to be kept in consideration while doing the reduction in transfer function that the behavior viz. peak overshoot, settling time, steady state error of the two systems (reduced and the original system) should be approximately same, so it is prime importance that the applied model order reduction technique should provide a more accurate approximation of original higher order system. The paper presents here the different categories of model order reduction techniques that can be applied to achieve the motto of model order reduction of higher order systems. The techniques presented are categorized into the four different categories to understand them and their merits and demerits and these will help in proper selection of the model order reduction technique to obtain the most accurate reduced order approximation of large scale system.

scholarly journals Research on improving measurement accuracy of acoustic transfer function of underwater vehicle

2021 ◽  
Vol 336 ◽  
pp. 01006
Author(s):  
Jiangqiao Li ◽  
Li Jiang ◽  
Fujian Yu ◽  
Ye Zhang ◽  
Kun Gao
Keyword(s):  
Transfer Function ◽  
Impulse Response ◽  
Time Domain ◽  
Transfer Functions ◽  
Least Squares Method ◽  
Signal To Noise Ratio ◽  
Random Noise ◽  
Phase Compensation ◽  
Acoustic Transfer Function ◽  
The Time Domain

To address the problem that acoustic transfer functions with underwater platforms cannot be measured accurately, this paper presents a method based on phase compensation to improve the accuracy of acoustic transfer function measurements on underwater platforms. The time-domain impulse response signals with multiple cycles are first collected and intercepted, and then their phase differences are estimated using the least-squares method, and phase compensation is used to align the phases of all the signals, and then the impulse response signals are weighted and averaged over all the impulse response signals to cancel out the random noise. The water pool test proves that this method reduces the measurement random noise while obtaining a high-fidelity time domain transfer function, which effectively improves the signal-to-noise ratio of the measurement. The method adopts only one measurement signal, and without changing the measurement system, the random noise is cancelled out by the in-phase superposition of the multi-cycle impulse response signals to avoid the nonlinear distortion of the measurement results.

METHOD FOR CALCULATION OF RADIO FREQUENCY INTERFERENCE BASED ON THE TRANSFER FUNCTION

2021 ◽  
pp. 50-55
Author(s):  
А.В. Башкиров ◽  
А.С. Демихова ◽  
Н.В. Астахов ◽  
М.В. Долженко ◽  
Д.Р. Елкин
Keyword(s):  
Transfer Function ◽  
Electromagnetic Interference ◽  
Transfer Functions ◽  
Least Squares Method ◽  
Noise Source ◽  
Dipole Moments ◽  
Radio Frequency Interference ◽  
Radio Interference ◽  
S Parameter ◽  
Two Stages

Предложен метод расчета передаточной функции для оценки задач помехоустойчивости и защищенности (RFI). Уравнения замкнутой формы аналитически выводятся из уравнений Максвелла и теоремы о взаимности. Задача RFI разложена на две части: дипольные моменты источника шума и передаточная функция связи с антенной. Передаточные функции могут быть получены либо из моделирования, либо из измерений. Простые измерения S-параметров могут помочь получить передаточные функции. Предложенный метод проверен с помощью численного моделирования и реальных экспериментов с использованием мобильного телефона. При моделировании источника помехи и связи помехи с антенной-приемником в предложенной работе задача разделяется на два этапа: прямая задача (источник шума излучает, а антенна выключена) и обратная задача (антенна возбуждается, а источник шума выключается). Инженеры могут использовать этот метод для выявления причин и устранения последствий воздействия электромагнитной помехи. Также предлагается метод расчета воздействия помех на основе передаточной функции для оценки степени искажения передаваемого сигнала. Данные уравнения позволяют четко разложить проблему радиопомех на две составляющие: источник шума и воздействие передаточной функции на антенну. В сравнении с обычным методом наименьших квадратов предлагаемый метод имеет лучшую точность (порядка 3 дБ) A method for calculating the transfer function for assessing noise immunity and security (RFI) is proposed. Closed-form equations are analytically derived from Maxwell's equations and the reciprocity theorem. The RFI problem is decomposed into two parts: the dipole moments of the noise source and the transfer function of coupling to the antenna. Transfer functions can be obtained either from simulations or measurements. Simple S-parameter measurements can provide transfer functions. The proposed method was verified using numerical simulation and real experiments using a mobile phone. When simulating the source of interference, and the communication of the interference with the antenna-receiver, in the proposed work the problem is divided into two stages: the direct problem (the noise source emits, and the antenna is turned off) and the inverse problem (the antenna is activated, and the noise source is turned off). Engineers can use this method to diagnose and correct the effects of electromagnetic interference. A method for calculating the effect of interference based on the transfer function is also proposed to assess the degree of distortion of the transmitted signal. These equations allow us to clearly decompose the problem of radio interference into two components: the noise source and the effect of the transfer function on the antenna. In comparison with the conventional least-squares method, the proposed method has better accuracy (about 3 dB)

scholarly journals Computational Algorithms for Reducing Rational Transfer Functions’ Order

2020 ◽  
Vol 19 (2) ◽  
pp. 330-356
Author(s):  
Mark Gourary ◽  
Sergey Rusakov ◽  
Mikhail Zharov ◽  
Sergey Ulyanov
Keyword(s):  
Transfer Function ◽  
Least Squares ◽  
Transfer Functions ◽  
Least Squares Method ◽  
Approximation Error ◽  
Iteration Step ◽  
Common Denominator ◽  
Transient Characteristics ◽  
Rational Transfer ◽  
Approximation Problems

A problem of reducing a linear time-invariant dynamic system is considered as a problem of approximating its initial rational transfer function with a similar function of a lower order. The initial transfer function  is also assumed to be rational. The approximation error is defined as the standard integral deviation of the transient characteristics of the initial and reduced transfer function in the time domain. The formulations of two main types of approximation problems are considered: a) the traditional problem of minimizing the approximation error at a given order of the reduced model; b) the proposed problem of minimizing the order of the model at  a given tolerance on the approximation error. Algorithms for solving approximation problems based on the Gauss-Newton iterative process are developed. At the iteration step, the current deviation of the transient characteristics is linearized with respect to the coefficients of the denominator of the reduced transfer function. Linearized deviations are used to obtain new values of the transfer function coefficients using the least-squares method  in a functional space based on Gram-Schmidt orthogonalization. The general form of expressions representing linearized deviations of transient characteristics is obtained. To solve the problem of minimizing the order of the transfer function in the framework of the least squares algorithm, the Gram-Schmidt process is also used. The completion criterion of the process is to achieve a given error tolerance. It is shown that the sequence of process steps corresponding to the alternation of coefficients of polynomials of the numerator and denominator of the transfer function provides the minimum order of transfer function. The paper presents an extension of the developed algorithms to the case of a vector transfer function with a common denominator. An algorithm is presented with the approximation error defined in the form of a geometric sum of scalar errors. The use of the minimax form for error estimation and the possibility of extending the proposed approach to the problem of reducing the irrational initial transfer function are discussed. Experimental code implementing the proposed algorithms is developed, and the results of numerical evaluations of test examples of various types are obtained.

Analytic integration of contrast transfer functions

1988 ◽  
Vol 46 ◽  
pp. 822-823
Author(s):  
Peter Rez
Keyword(s):  
Wave Function ◽  
Transfer Function ◽  
Transfer Functions ◽  
Spherical Aberration ◽  
Continuous Distribution ◽  
Objective Lens ◽  
Direction Parallel ◽  
Scattering Angle ◽  
Analytic Integration ◽  
Image Position

In high resolution microscopy the image amplitude is given by the convolution of the specimen exit surface wave function and the microscope objective lens transfer function. This is usually done by multiplying the wave function and the transfer function in reciprocal space and integrating over the effective aperture. For very thin specimens the scattering can be represented by a weak phase object and the amplitude observed in the image plane is1where fe (Θ) is the electron scattering factor, r is a postition variable, Θ a scattering angle and x(Θ) the lens transfer function. x(Θ) is given by2where Cs is the objective lens spherical aberration coefficient, the wavelength, and f the defocus.We shall consider one dimensional scattering that might arise from a cross sectional specimen containing disordered planes of a heavy element stacked in a regular sequence among planes of lighter elements. In a direction parallel to the disordered planes there will be a continuous distribution of scattering angle.

Trombone Transfer Functions: Comparison Between Frequency-Swept Sine Wave and Human Performer Input

2012 ◽  
Vol 37 (4) ◽  
pp. 447-454
Author(s):  
James W. Beauchamp
Keyword(s):  
Transfer Function ◽  
Transfer Functions ◽  
Cutoff Frequency ◽  
Sine Wave ◽  
Nonlinear Propagation ◽  
Linear Filter ◽  
Wave System ◽  
General Effect ◽  
Filter Characteristic ◽  
High Pass

Abstract Source/filter models have frequently been used to model sound production of the vocal apparatus and musical instruments. Beginning in 1968, in an effort to measure the transfer function (i.e., transmission response or filter characteristic) of a trombone while being played by expert musicians, sound pressure signals from the mouthpiece and the trombone bell output were recorded in an anechoic room and then subjected to harmonic spectrum analysis. Output/input ratios of the signals’ harmonic amplitudes plotted vs. harmonic frequency then became points on the trombone’s transfer function. The first such recordings were made on analog 1/4 inch stereo magnetic tape. In 2000 digital recordings of trombone mouthpiece and anechoic output signals were made that provide a more accurate measurement of the trombone filter characteristic. Results show that the filter is a high-pass type with a cutoff frequency around 1000 Hz. Whereas the characteristic below cutoff is quite stable, above cutoff it is extremely variable, depending on level. In addition, measurements made using a swept-sine-wave system in 1972 verified the high-pass behavior, but they also showed a series of resonances whose minima correspond to the harmonic frequencies which occur under performance conditions. For frequencies below cutoff the two types of measurements corresponded well, but above cutoff there was a considerable difference. The general effect is that output harmonics above cutoff are greater than would be expected from linear filter theory, and this effect becomes stronger as input pressure increases. In the 1990s and early 2000s this nonlinear effect was verified by theory and measurements which showed that nonlinear propagation takes place in the trombone, causing a wave steepening effect at high amplitudes, thus increasing the relative strengths of the upper harmonics.

Structural-parametric model of an electroelastic actuator for nanomechanics

2020 ◽  
pp. 3-11
Author(s):  
S.M. Afonin
Keyword(s):  
Transfer Function ◽  
Piezoelectric Actuator ◽  
Transfer Functions ◽  
Piezoelectric Effect ◽  
Parametric Model ◽  
Elastic Compliance ◽  
Parametric Models ◽  
Piezo Actuator ◽  
Structural Scheme ◽  
Model Transfer

Structural-parametric models, structural schemes are constructed and the transfer functions of electro-elastic actuators for nanomechanics are determined. The transfer functions of the piezoelectric actuator with the generalized piezoelectric effect are obtained. The changes in the elastic compliance and rigidity of the piezoactuator are determined taking into account the type of control. Keywords electro-elastic actuator, piezo actuator, structural-parametric model, transfer function, parametric structural scheme

scholarly journals Determination of the Optimal Neural Network Transfer Function for Response Surface Methodology and Robust Design

2021 ◽  
Vol 11 (15) ◽  
pp. 6768
Author(s):  
Tuan-Ho Le ◽  
Hyeonae Jang ◽  
Sangmun Shin
Keyword(s):  
Response Surface Methodology ◽  
Transfer Function ◽  
Response Surface ◽  
Robust Design ◽  
Transfer Functions ◽  
Desirability Function ◽  
Likelihood Estimation ◽  
Statistical Simulation ◽  
Screening Procedure ◽  
Function Estimation

Response surface methodology (RSM) has been widely recognized as an essential estimation tool in many robust design studies investigating the second-order polynomial functional relationship between the responses of interest and their associated input variables. However, there is scope for improvement in the flexibility of estimation models and the accuracy of their results. Although many NN-based estimations and optimization approaches have been reported in the literature, a closed functional form is not readily available. To address this limitation, a maximum-likelihood estimation approach for an NN-based response function estimation (NRFE) is used to obtain the functional forms of the process mean and standard deviation. While the estimation results of most existing NN-based approaches depend primarily on their transfer functions, this approach often requires a screening procedure for various transfer functions. In this study, the proposed NRFE identifies a new screening procedure to obtain the best transfer function in an NN structure using a desirability function family while determining its associated weight parameters. A statistical simulation was performed to evaluate the efficiency of the proposed NRFE method. In this particular simulation, the proposed NRFE method provided significantly better results than conventional RSM. Finally, a numerical example is used for validating the proposed method.

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